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Here's a quick how-to guide on playing the zombie class.
Keep in mind that as the event goes on, some details might change.

Step 1: Infection
The following things will get you infected:
- Killing infected critters (roaches) in major cities
- Clicking on Conspicuous Crates (green glowing boxes) in major cities
- Being attacked by Plagued Zombies
- Being blown up by Plagued Zombies (Zombie Explosion)
- Standing in the plague cloud created by Plagued Zombies for too long (Retch)

Step 2: Incubation (currently 5 minutes)
Several options for the incubation process:
- Spend the next 5 minutes not getting cleansed by players and staying well away from Argent Dawn healers located in major cities.
- Have a Plagued Zombie use their Mangle ability on you. It will periodically reduced your infection timer by 30 seconds.
- Use an immunity ability on yourself to instantly become a zombie (Divine Shield, Ice Block confirmed to work).

Step 3: Plagued Zombie!
Now the periphery of your screen should glow red as you're transformed into one of the living dead!
Your main action bar will be replaced with a new zombie action bar with several zombie abilities (discussed below).

You will slowly lose health, so you must keep attacking the living to regenerate your health!
You will also move fairly slowly (walking speed) unless you're lurching.

While in this state, you are part of the scourge faction and is now hostile to all players and most NPCs.
You can also converse and trade with other player zombies regardless of "former" faction.
Regular Scourge zombies will now be friendly to you (confirmed in EPL, and Stratholme).

Once you're a zombie, the only way to stop being one is to become even more deader (getting killed).
I would suggest taking off your armor, as getting killed by NPCs still cause durability damage, and armor stats do not improve your zombie stats.

Rogues can also stealth if they become zombies (unconfirmed, but lots of similar stories).

Zombie Skills!
Your main objective as a Plague Zombie is to infect as many of the living as possible!
To help you, there're several skills.

1. Mangle: Your main damage skill on 3 second cooldown. This also reduces a player or NPC's infection timer. As long as you keep mangling them, the timer will periodically decrease by 30 seconds. This also puts an armor debuff on the target, but so far it doesn't seem to do much.

2. Retch: Your short range AOE disease cloud that heals zombies and damages everything else. If humanoid NPCs and players don't get out of the cloud fast enough, they'll become infected as well. If you have multiple player zombies, this is a skill you should use on every cooldown, and it stacks. Alternatively, if you're traveling in player groups, rotating retches can help slow the drain on your health.

3. Beckoning Groan: Use this to call nearby non-player zombies to your location. They will follow for about 10-15 seconds. Up to 4 targets affected, but who gets affected is random and not particularly reliable at times.

4. Lurch: Gives all nearby zombies a temporary speed boost to chase down living humans. I would use this in conjunction with Beckoning Groan to keep everyone together. This should probably be put on round-robin.

5. Zombie Explosion: After channeling for 4 seconds (doesn't seem to have spell pushback), you explode. This causes NO damage but instantly infects everyone around you. If the living has superior numbers, use this quick to infect them all! (unconfirmed: doesn't damage durability)